55.12.9 problem 46

Internal problem ID [13440]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-4. Equations with cotangent.
Problem number : 46
Date solved : Wednesday, October 01, 2025 at 12:12:22 PM
CAS classification : [_Riccati]

\begin{align*} \left (a \cot \left (\lambda x \right )+b \right ) y^{\prime }&=y^{2}+c \cot \left (\mu x \right ) y-d^{2}+c d \cot \left (\mu x \right ) \end{align*}

✓ Maple. Time used: 0.014 (sec). Leaf size: 366

ode:=(a*cot(lambda*x)+b)*diff(y(x),x) = y(x)^2+c*cot(x*mu)*y(x)-d^2+c*d*cot(x*mu); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-{\mathrm e}^{\frac {c \int \frac {\cot \left (\mu x \right )}{a \cot \left (\lambda x \right )+b}d x \lambda \left (a^{2}+b^{2}\right )-2 b \left (\operatorname {arccot}\left (\cot \left (\lambda x \right )\right )-\frac {\pi }{2}\right ) d}{\lambda \left (a^{2}+b^{2}\right )}} \left (\csc \left (\lambda x \right )^{2}\right )^{-\frac {d a}{\lambda \left (a^{2}+b^{2}\right )}} \left (a \cot \left (\lambda x \right )+b \right )^{\frac {2 d a}{\lambda \left (a^{2}+b^{2}\right )}}-\left (\int \left (a \cot \left (\lambda x \right )+b \right )^{\frac {\left (-a^{2}-b^{2}\right ) \lambda +2 a d}{\lambda \left (a^{2}+b^{2}\right )}} {\mathrm e}^{\frac {c \int \frac {\cot \left (\mu x \right )}{a \cot \left (\lambda x \right )+b}d x \lambda \left (a^{2}+b^{2}\right )-2 b \left (\operatorname {arccot}\left (\cot \left (\lambda x \right )\right )-\frac {\pi }{2}\right ) d}{\lambda \left (a^{2}+b^{2}\right )}} \left (\csc \left (\lambda x \right )^{2}\right )^{-\frac {d a}{\lambda \left (a^{2}+b^{2}\right )}}d x -c_1 \right ) d}{\int \left (a \cot \left (\lambda x \right )+b \right )^{\frac {\left (-a^{2}-b^{2}\right ) \lambda +2 a d}{\lambda \left (a^{2}+b^{2}\right )}} {\mathrm e}^{\frac {c \int \frac {\cot \left (\mu x \right )}{a \cot \left (\lambda x \right )+b}d x \lambda \left (a^{2}+b^{2}\right )-2 b \left (\operatorname {arccot}\left (\cot \left (\lambda x \right )\right )-\frac {\pi }{2}\right ) d}{\lambda \left (a^{2}+b^{2}\right )}} \left (\csc \left (\lambda x \right )^{2}\right )^{-\frac {d a}{\lambda \left (a^{2}+b^{2}\right )}}d x -c_1} \]

✓ Mathematica. Time used: 27.35 (sec). Leaf size: 799

ode=(a*Cot[\[Lambda]*x]+b)*D[y[x],x]==y[x]^2+c*Cot[\[Mu]*x]*y[x]-d^2+c*d*Cot[\[Mu]*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
d = symbols("d") 
lambda_ = symbols("lambda_") 
mu = symbols("mu") 
y = Function("y") 
ode = Eq(-c*d/tan(mu*x) - c*y(x)/tan(mu*x) + d**2 + (a/tan(lambda_*x) + b)*Derivative(y(x), x) - y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

TypeError : Invalid NaN comparison